# 0.3 Signal processing in processing: sampling and quantization  (Page 3/4)

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## 2-d: images

Let us assume we have a continuous distribution, on a plane, of values of luminance or, more simply stated, animage. In order to process it using a computer we have to reduce it to a sequence of numbers by means ofsampling. There are several ways to sample an image, or read its values of luminance at discrete points. Thesimplest way is to use a regular grid, with spatial steps $X$ e $Y$ . Similarly to what we did for sounds, we define the spatial sampling rates ${F}_{X}=\frac{1}{X}$ and ${F}_{Y}=\frac{1}{Y}$ . As in the one-dimensional case, also for two-dimensional signals, or images, sampling can bedescribed by three facts and a theorem.

• The Fourier Transform of a discrete-space signal is a function (called spectrum ) of two continuous variables ${\omega }_{X}$ and ${\omega }_{Y}$ , and it is periodic in two dimensions with periods $2\pi$ . Given a couple of values ${\omega }_{X}$ and ${\omega }_{Y}$ , the Fourier transform gives back a complex number that can be interpreted as magnitude andphase (translation in space) of the sinusoidal component at such spatial frequencies.
• Sampling the continuous-space signal $s(x, y)$ with the regular grid of steps $X$ , $Y$ , gives a discrete-space signal $s(m, n)=s(mX, nY)$ , which is a function of the discrete variables $m$ and $n$ .
• Sampling a continuous-space signal with spatial frequencies ${F}_{X}$ and ${F}_{Y}$ gives a discrete-space signal whose spectrum is the periodic replication along the grid of steps ${F}_{X}$ and ${F}_{Y}$ of the original signal spectrum. The Fourier variables ${\omega }_{X}$ and ${\omega }_{Y}$ correspond to the frequencies (in cycles per meter) represented by the variables ${f}_{X}=\frac{{\omega }_{X}}{2\pi X}$ and ${f}_{Y}=\frac{{\omega }_{Y}}{2\pi Y}$ .

The [link] shows an example of spectrum of a two-dimensional sampled signal. There, thecontinuous-space signal had all and only the frequency components included in the central hexagon. The hexagonalshape of the spectral support (region of non-null spectral energy) is merely illustrative. The replicas of the originalspectrum are often called spectral images .

Given the above facts , we can have an intuitive understanding of the Sampling Theorem.

## Sampling theorem (in 2d)

A continuous-space signal $s(x, y)$ , whose spectral content is limited to spatial frequencies belonging to the rectangle having semi-edges ${F}_{bX}$ and ${F}_{bY}$ (i.e., bandlimited) can be recovered from its sampled version $s(m, n)$ if the spatial sampling rates are larger than twice the respective bandwidths (i.e., if ${F}_{X}> 2{F}_{bX}$ and ${F}_{Y}> 2{F}_{bY}$ )

In practice, the spatial sampling step can not be larger than the semi-period of the finest spatial frequency (or thefinest detail) that is represented in the image. The reconstruction can only be done through a filter thateliminates all the spectral images but the one coming directly from the original continuous-space signal. In otherwords, the filter will cut all images whose frequency components are higher than the Nyquist frequency defined as $\frac{{F}_{X}}{2}$ and $\frac{{F}_{Y}}{2}$ along the two axes. The condition required by the sampling theorem is equivalent to requiring that there are no overlaps between spectral images. If there were suchoverlaps, it wouldn't be possible to eliminate the copies of the original signal spectrum by means of filtering. In case of overlapping, a filtercutting all frequency components higher than the Nyquist frequency would give back a signal that is affected byaliasing.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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